A local asymptotic expansion for a local solution of the Stokes system

TL;DR

This paper establishes a local asymptotic expansion for solutions of the Stokes system near points where the velocity vanishes, providing polynomial approximations with controlled error and extending results to related fluid dynamics systems.

Contribution

It introduces a divergence-free polynomial approximation for solutions of the Stokes system with a specific order of vanishing, extending to Oseen and Navier-Stokes equations.

Findings

Existence of polynomial approximations with order $d+ ext{alpha}$

Approximation satisfies a Stokes equation with polynomial forcing

Results extend to Oseen and Navier-Stokes systems

Abstract

We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$. The results extend to Oseen systems and to the Navier-Stokes equation.